The number e first comes into mathematics in a very minor way. This was in 1618 when, in an appendix to Napier's work on logarithms, a table appeared giving the natural logarithms of various numbers

1

• The number e first comes into mathematics in a
  very minor way. This was in 1618 when, in an
  appendix to Napier's work on logarithms, a table
  appeared giving the natural logarithms of various
  numbers

2

• The next possible occurrence of e is again
  dubious. In 1647 Saint-Vincent computed the area
  under a rectangular hyperbola. Whether he
  recognised the connection with logarithms is open
  to debate, and even if he did there was little
  reason for him to come across the number e
  explicitly.
• Certainly by 1661 Huygens understood the relation
  between the rectangular hyperbola and the
  logarithm. He examined explicitly the relation
  between the area under the rectangular hyperbola
  yx 1 and the logarithm. Of course, the number e
  is such that the area under the rectangular
  hyperbola from 1 to e is equal to 1. This is the
  property that makes e the base of natural
  logarithms, but this was not understood by
  mathematicians at this time, although they were
  slowly approaching such an understanding.

3

• In 1683 Jacob Bernoulli looked at the problem of
  compound interest and, in examining continuous
  compound interest, he tried to find the limit of
  (1 1/n)n as n tends to infinity. He used the
  binomial theorem to show that the limit had to
  lie between 2 and 3 so we could consider this to
  be the first approximation found to e. Also if we
  accept this as a definition of e, it is the first
  time that a number was defined by a limiting
  process. He certainly did not recognise any
  connection between his work and that on
  logarithms.

4

• As far as we know the first time the number e
  appears in its own right is in 1690. In that year
  Leibniz wrote a letter to Huygens and in this he
  used the notation b for what we now call e. At
  last the number e had a name (even if not its
  present one) and it was recognised.

5

• It would be fair to say that Johann Bernoulli
  began the study of the calculus of the
  exponential function in 1697 when he published
  Principia calculi exponentialium seu
  percurrentium. The work involves the calculation
  of various exponential series and many results
  are achieved with term by term integration.

6

• So much of our mathematical notation is due to
  Euler that it will come as no surprise to find
  that the notation e for this number is due to
  him. The claim which has sometimes been made,
  however, that Euler used the letter e because it
  was the first letter of his name is ridiculous.
  It is probably not even the case that the e comes
  from "exponential", but it may have just be the
  next vowel after "a" and Euler was already using
  the notation "a" in his work. Whatever the
  reason, the notation e made its first appearance
  in a letter Euler wrote to Goldbach in 1731.

7

• He made various discoveries regarding e in the
  following years, but it was not until 1748 when
  Euler published Introductio in Analysin
  infinitorum that he gave a full treatment of the
  ideas surrounding e. He showed that
• e 1 1/1! 1/2! 1/3! ...
• and that e is the limit of (1 1/n)n as n tends
  to infinity. Euler gave an approximation for e to
  18 decimal places,
• e 2.718281828459045235
• without saying where this came from. It is
  likely that he calculated the value himself, but
  if so there is no indication of how this was
  done. In fact taking about 20 terms of 1 1/1!
  1/2! 1/3! ... will give the accuracy which
  Euler gave.